# Declination as a Metric to Detect Partisan Gerrymandering

## TL;DR

This paper investigates the Declination metric for detecting partisan gerrymandering, analyzing its effectiveness under various turnout scenarios and comparing it to the Efficiency Gap, highlighting its limitations and differences.

## Contribution

The paper provides a detailed mathematical analysis of the Declination metric, identifying when it equals zero and comparing its behavior to the Efficiency Gap in detecting gerrymandering.

## Key findings

- Declination equals zero for specific vote-seat share pairs under certain turnout conditions.
- Declination cannot detect all forms of packing and cracking gerrymandering.
- Declination and Efficiency Gap can behave differently in various scenarios.

## Abstract

We explore the Declination, a new metric intended to detect partisan gerrymandering. We consider instances in which each district has equal turnout, the maximum turnout to minimum turnout is bounded, and turnout is unrestricted. For each of these cases, we show exactly which vote-share, seat-share pairs $(V,S)$ have an election outcome with Declination equal to 0. We also show how our analyses can be applied to finding vote-share, seat-share pairs that are possible for nonzero Declination. Within our analyses, we show that Declination cannot detect all forms of packing and cracking, and we compare the Declination to the Efficiency Gap. We show that these two metrics can behave quite differently, and give explicit examples of that occurring.